Group #16846544 — subgroups

Subgroups — #16846544

Order 3968.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521073	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11758425	C₂	1	4	yes	maximal, complemented
3	#14520895	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#41×C₃₁	C₂²	1	2	yes	—
6	C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	C₃₁×#127	C₄	1	1	yes	characteristic, complemented
8	C₃₁×#124	C₄	1	2	yes	complemented
9	C₃₁×#121	—	2	2	no	—
10	C₄×D₄×C₃₁	—	2	2	no	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
13	C₂×#92474	C₂²	1	2	yes	—
14	#11758425	—	2	8	no	—
15	C₃₁⋊#95	—	2	2	no	—
16	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
17	#41×C₃₁	—	2	4	no	—
18	#41×C₃₁	—	2	4	no	—
19	C₂×Q₈×C₃₁	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₃₁	—	2	2	no	—
21	C₂×D₄×C₃₁	—	2	2	no	—
22	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₁	—	2	2	no	—
24	C₂²×C₄×C₃₁	D₄	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₁	—	2	2	no	—
27	C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂²⋊C₄×C₃₁	—	2	4	no	—
29	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₃₁	D₄	1	2	yes	—
32	C₄×Dic₃₁	—	2	2	no	—
33	C₄×Dic₃₁	D₄	1	4	yes	complemented
34	#92474	—	2	4	no	—
35	C₂×C₃₁⋊C₈	—	2	2	no	—
36	C₂³×C₃₁	—	2	2	no	—
37	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₃₁	—	2	4	no	—
39	D₄×C₃₁	—	2	4	no	—
40	D₄×C₃₁	—	4	4	no	—
41	D₄×C₃₁	—	4	4	no	—
42	C₂×C₄×C₃₁	—	2	2	no	—
43	C₂×C₄×C₃₁	—	4	2	no	—
44	C₂×C₄×C₃₁	—	4	2	no	—
45	C₂×C₄×C₃₁	—	4	2	no	—
46	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	#41	1	2	yes	—
52	C₂×Dic₃₁	—	2	4	no	—
53	C₂×Dic₃₁	—	4	4	no	—
54	C₃₁⋊C₈	—	4	2	no	—
55	#1963	—	31	1	no	maximal, Sylow
56	C₂²×C₃₁	—	2	2	no	—
57	C₂²×C₃₁	—	4	2	no	—
58	C₂²×C₃₁	—	4	4	no	—
59	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₃₁	#121	1	2	yes	—
61	C₄×C₃₁	—	2	2	no	—
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	—	4	2	no	—
64	C₄×C₃₁	C₄×D₄	1	2	yes	—
65	C₄×C₃₁	#124	1	1	yes	characteristic
66	C₄×C₃₁	#127	1	1	yes	characteristic
67	Dic₃₁	—	4	4	no	—
68	#532	D₃₁	1	1	yes	characteristic
69	#423	—	31	1	no	—
70	C₂×#104	—	31	4	no	—
71	#376	—	31	1	no	—
72	C₂×C₃₁	—	2	2	no	—
73	C₂×C₃₁	—	8	2	no	—
74	C₂×C₃₁	#390	1	1	yes	characteristic
75	C₂×C₃₁	#421	1	2	yes	—
76	C₂×#41	C₂×D₃₁	1	2	yes	—
77	C₂×M₁₆	—	31	2	no	—
78	C₄⋊Q₈	Dic₃₁	1	1	yes	characteristic, complemented
79	#127	Dic₃₁	1	1	yes	characteristic, complemented
80	#124	Dic₃₁	1	2	yes	complemented
81	#121	—	2	2	no	—
82	C₄×D₄	—	2	2	no	—
83	C₂×C₄²	—	31	2	no	—
84	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
85	C₄⋊C₈	—	62	2	no	—
86	#104	—	62	8	no	—
87	#95	—	62	2	no	—
88	C₃₁	#1963	1	1	yes	characteristic, complemented, Sylow
89	#41	—	2	4	no	—
90	#41	—	2	4	no	—
91	C₂×Q₈	C₂×Dic₃₁	1	2	yes	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×Dic₃₁	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	62	2	no	—
97	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
98	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
99	M₁₆	—	62	4	no	—
100	C₂×C₈	—	62	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	—	31	4	no	—
104	C₄²	—	62	2	no	—
105	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
106	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
107	C₄²	C₂²⋊D₃₁	1	2	yes	—
108	C₂³	—	2	2	no	—
109	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
110	Q₈	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	4	no	—
113	D₄	—	4	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	62	4	no	—
119	C₂×C₄	—	124	4	no	—
120	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
121	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
122	C₂×C₄	D₄×D₃₁	1	2	yes	—
123	C₂×C₄	#92497	1	2	yes	—
124	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
126	C₈	—	124	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	#11758515	1	2	yes	—
131	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	2	no	—
135	C₄	—	124	4	no	—
136	C₄	D₄×Dic₃₁	1	2	yes	—
137	C₄	#11758513	1	1	yes	characteristic
138	C₄	#11758517	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	8	2	no	—
141	C₂	#14520952	1	1	yes	characteristic
142	C₂	#14521066	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521073

C₂

yes

maximal, characteristic, complemented

C₂×#11758425

C₂

yes

maximal, complemented

#14520895

C₂

yes

maximal, characteristic, complemented

C₃₁×#532

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

C₂²

yes

—

C₃₁×C₄⋊Q₈

C₄

yes

characteristic, complemented

C₃₁×#127

C₄

yes

characteristic, complemented

C₃₁×#124

C₄

yes

complemented

C₃₁×#121

—

C₄×D₄×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂×#92474

C₂²

yes

—

#11758425

—

C₃₁⋊#95

—

C₄⋊(C₃₁⋊C₈)

—

#41×C₃₁

—

#41×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

complemented

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#1963

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

#121

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

#532

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#390

yes

characteristic

C₂×C₃₁

#421

yes

—

C₂×#41

C₂×D₃₁

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₃₁

yes

characteristic, complemented

#127

Dic₃₁

yes

characteristic, complemented

#124

Dic₃₁

yes

complemented

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

#104

—

#95

—

C₃₁

#1963

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

C₂×Dic₃₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂×Dic₃₁

yes

characteristic

106

C₄²

C₂×Dic₃₁

yes

characteristic

107

C₄²

C₂²⋊D₃₁

yes

—

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

124

—

120

C₂×C₄

C₂²⋊Dic₃₁

yes

—

121

C₂×C₄

C₂²⋊Dic₃₁

yes

—

122

C₂×C₄

D₄×D₃₁

yes

—

123

C₂×C₄

#92497

yes

—

124

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

126

C₈

—

124

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

#11758515

yes

—

131

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

124

—

136

C₄

D₄×Dic₃₁

yes

—

137

C₄

#11758513

yes

characteristic

138

C₄

#11758517

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

#14520952

yes

characteristic

142

C₂

#14521066

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting, prime radical	C₃₁×#532	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#532	D₃₁
hypocenter	C₃₁	#1963
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting, prime radical

C₃₁×#532

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #14521073	C₂	semidirect
2. C₂×#11758425	C₂	semidirect
3. #14520895	C₂	semidirect
6. C₃₁×C₄⋊Q₈	C₄	semidirect
7. C₃₁×#127	C₄	semidirect
8. C₃₁×#124	C₄	semidirect
33. C₄×Dic₃₁	D₄	semidirect
78. C₄⋊Q₈	Dic₃₁	semidirect
79. #127	Dic₃₁	semidirect
80. #124	Dic₃₁	semidirect
88. C₃₁	#1963	semidirect

H ≅ G/N

type

1. #14521073

C₂

semidirect

2. C₂×#11758425

C₂

semidirect

3. #14520895

C₂

semidirect

6. C₃₁×C₄⋊Q₈

C₄

semidirect

7. C₃₁×#127

C₄

semidirect

8. C₃₁×#124

C₄

semidirect

33. C₄×Dic₃₁

D₄

semidirect

78. C₄⋊Q₈

Dic₃₁

semidirect

79. #127

Dic₃₁

semidirect

80. #124

Dic₃₁

semidirect

88. C₃₁

#1963

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14521073	2
C₂×#11758425	2
#14520895	2
C₃₁×#532	2
#1963	31

subgroup

index

C₃₁×#532

Subgroups — #16846544

Conjugacy classes of subgroups

Complements

Maximal subgroups